In an orthogonal set of sequences the inner product between any two sequences from the set is zero. A quasi-orthogonal (QO) set of sequences is a union of a number of different orthogonal sets of sequences, where all the sequences are of the same length and energy, such that the absolute value of the inner product of any two sequences belonging to the different orthogonal sets is much less than the energy of the sequences.
The interest in QO sets of polyphase sequences has been increased recently due to extensive use of Zadoff-Chu (ZC) sequences in various parts of LTE cellular standard. A ZC sequence of length N and of root index u is defined as:zu(k)=WNuk(k+N mod2+2q)/2,k=0,1, . . . ,N−1,  (1)where WN=e−j2π/N, j=√{square root over (−1)}, while u, N are any positive integers such that u<N, (u,N)=1, while q is any integer.
In particular, LTE Random Access Channel (RACH) preambles, as well as uplink Sounding Reference Signals (SRS), are defined as cyclic versions of a prime length ZC sequence with cell-specific root indices. As a ZC sequence is a Constant Amplitude Zero (periodic) Autocorrelation (CAZAC) sequence, the set of all its cyclic versions form an orthogonal set of sequences. The union of N−1 cell-specific orthogonal sets of cyclic prime length ZC sequences, corresponding to all N−1 different root indices, forms a QO set of polyphase sequences {sm,l(k)}, m=1, . . . , N−1, l,k=0, 1, . . . , N−1, N is a prime, which can be described mathematically as:sm,l(k)=zm(k+l(mod N)),  (2)m=1, . . . , N−1, l,k=0, 1, . . . , N−1.
The usual measure of interference between the synchronization sequences belonging to different orthogonal sets is the inner product of the sequences. The inner product of two sequences x={x(k)} and y={y(k)}, k=0, 1, . . . , N−1, is defined as:
                                          〈                          x              ,              y                        〉                    =                                    ∑                              k                =                0                                            N                -                1                                      ⁢                                                            x                  *                                ⁡                                  (                  k                  )                                            ⁢                              y                ⁡                                  (                  k                  )                                                                    ,                            a        .            where “*” denotes complex conjugation.
The absolute value of the inner product of any two sequences (2) belonging to the different orthogonal sets is equal to √{square root over (N)}. This property of the inner product follows directly from the cross-correlation properties of ZC sequences.
The construction of QO sets has been initially motivated by the need to create the user-specific sets of orthogonal spreading codes, used both for information error-control coding and spectrum spreading in synchronous code-division multiple-access (CDMA) systems, so that the interference in the system is minimized. Another motivation was the need to increase the number of users in a synchronous CDMA system using Walsh sequences, such as the downlink of IS-95 cellular system, without increasing the length of spreading sequences.
The QO sets are typically constructed by using specially designed cover sequences, also called “mask sequences” or “signature sequences”, so that all the sequences within an orthogonal set are multiplied symbol-by-symbol with a common unique cover/signature sequence. Minimization of the maximum absolute inner product between any two sequences belonging to the different orthogonal sets is the primary criterion for the signature design, although some additional criteria have been considered as well.
As the design of QO sets has been always motivated by the applications in the real-life cellular systems, the resulting QO sequences were typically either binary or quadriphase, matched to the modulation formats used in real systems prior to LTE systems.
The QO sets of polyphase sequences have been introduced in LTE system, as mentioned before. An another construction of QO sets of polyphase sequences has been proposed based on a modification of prime-length Alltop's cubic phase sequences. The QO sets of cubic phase sequences of prime length N are defined as:sm,l(k)=ej2π(k-m)3/Nej2πlk/N,m,l,k=0,1, . . . , N−1,  (3)
The first exponential term, the m-th cyclically shifted version of the cubic phase sequence, is the mask sequence defining the m-th orthogonal set. Indeed, the construction of equation (3) produces N orthogonal subsets, each containing N sequences of length N, such that the absolute value of the inner product of any two sequences belonging to the different orthogonal sets is less than or equal to √{square root over (N)}. Similar construction has also appeared as:
                    s                  m          ,          l                    ⁡              (        k        )              =                  1                  N                    ⁢              ⅇ                  j          ⁢                                          ⁢          2          ⁢                                    π              ⁡                              (                                                      k                    3                                    -                  m                                )                                      /            N                              ⁢              ⅇ                  j          ⁢                                          ⁢          2          ⁢          π          ⁢                                          ⁢          l          ⁢                                          ⁢                      k            /            N                                ,  m  ,  l  ,      k    =    0    ,  1  ,  …  ⁢          ,      N    -    1    ,
where the sequence elements are normalized with 1/√{square root over (N)} to have all sequences of unit energy. The above equation seems to be just a version of equation (3) with a typo, as it is trivial to show that it does not generate a QO set of sequences (the different orthogonal subsets are equivalent versions of each other, up to a complex constant).
Expected increase of the number of users in the future versions of wireless cellular communication systems, such as the LTE cellular system, will demand an increased number of RACH preambles and SRSs having minimal mutual multiple access interference at the common serving base station receiver. Thus it might be needed to have QO sets with more than N orthogonal subsets of sequences of length N, where the maximum absolute value of the inner product is kept much lower than the sequence energy.
Additionally, it might be desirable that some special additional conditions are satisfied, particularly to be possible to create QO sets that have minimum interference between a certain number of orthogonal sets.